TAILIEUCHUNG - Ebook A first course in the finite element method: Part 2
(BQ) Part 2 book "A first course in the finite element method" has contents: Axisymmetric elements, isoparametric formulation, three dimensional stress analysis, plate bending element, heat transfer and mass transport, fluid flow, thermal stress, structural dynamics and time-dependent heat transfer. | CHAPTER 9 Axisymmetric Elements Introduction In previous chapters, we have been concerned with line or one-dimensional elements (Chapters 2–5) and two-dimensional elements (Chapters 6–8). In this chapter, we consider a special two-dimensional element called the axisymmetric element. This element is quite useful when symmetry with respect to geometry and loading exists about an axis of the body being analyzed. Problems that involve soil masses subjected to circular footing loads or thick-walled pressure vessels can often be analyzed using the element developed in this chapter. We begin with the development of the stiffness matrix for the simplest axisymmetric element, the triangular torus, whose vertical cross section is a plane triangle. We then present the longhand solution of a thick-walled pressure vessel to illustrate the use of the axisymmetric element equations. This is followed by a description of some typical large-scale problems that have been modeled using the axisymmetric element. d Derivation of the Stiffness Matrix d In this section, we will derive the stiffness matrix and the body and surface force matrices for the axisymmetric element. However, before the development, we will first present some fundamental concepts prerequisite to the understanding of the derivation. Axisymmetric elements are triangular tori such that each element is symmetric with respect to geometry and loading about an axis such as the z axis in Figure 9–1. Hence, the z axis is called the axis of symmetry or the axis of revolution. Each vertical cross section of the element is a plane triangle. The nodal points of an axisymmetric triangular element describe circumferential lines, as indicated in Figure 9–1. In plane stress problems, stresses exist only in the x-y plane. In axisymmetric problems, the radial displacements develop circumferential strains that induce stresses sr , sy , sz , and trz , where r, y, and z indicate the radial, circumferential, and .
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